I may have discovered an explanation to the Zeno Paradox. However, in this explanation, the concept of a race between to objects at different speeds is simplified into one object traveling a given distance. The distance traveled in this example can be represented by the variable D. If the distance traveled were divided into an infinite number of segments, then each segment could be represented by D/(infinite). Therefore, if each of these D/(infinite) segments were added together an infinite amount of times since there is an infinite amount of divisions then if would be D/(infinite) * infinite which would equal D.

The same is true with time, and as the number of divisions goes to infinite the time to cross those divisions goes to 1/(infinite) then it is easy to resolve the paradox.

Hurkyl
Staff Emeritus
Gold Member
I may have discovered an explanation to the Zeno Paradox.
Then first, you need to precisely state Zeno's paradox. There are many different ways to interpret it. Some interpretations are extremely easy to refute, due to an obvious error in the argument. I've seen others interpret Zeno's paradox as one of the first arguments (by reductio ad absurdum) that that error really is an error. (It probably wasn't so obvious back then)

If the distance traveled were divided into an infinite number of segments,
What segments?

then each segment could be represented by D/(infinite).
What do you mean by (infinite)? And why do you think you can divide by it? And how would the result represent a segment?

Therefore, if each of these D/(infinite) segments were added together an infinite amount of times
You can't add an infinite number of times. What operation are you really intending to use? (e.g. the basic definition of infinite sum from calculus?)

since there is an infinite amount of divisions
What is a division? And why would there be infinitely many of them?

then if would be D/(infinite) * infinite
What does 'infinite' mean here? And why do you think you can multiply D/(infinite) by it?

which would equal D.
And why would you think it gives this result?

DaveC426913
Gold Member
I may have discovered an explanation to the Zeno Paradox. However, in this explanation, the concept of a race between to objects at different speeds is simplified into one object traveling a given distance. The distance traveled in this example can be represented by the variable D. If the distance traveled were divided into an infinite number of segments, then each segment could be represented by D/(infinite). Therefore, if each of these D/(infinite) segments were added together an infinite amount of times since there is an infinite amount of divisions then if would be D/(infinite) * infinite which would equal D.

The same is true with time, and as the number of divisions goes to infinite the time to cross those divisions goes to 1/(infinite) then it is easy to resolve the paradox.

D/infinity*infinity does NOT result in D; it results in undefined - which means 'this has no unique answer'.

What segments?
The segments that the line is divided into. If we have a line of length D and we divide it into an infinite amount of sections then each section would be the width of D/(infinite)

To answer another of your questions right off the bat, (infinite) is equal to infinite just put the parenthesis to be helpful.

Would you like me to define every word of my post to you?

What is a division? And why would there be infinitely many of them?
A division is: 1. the act or process of dividing; state of being divided.
Dividing is: 1. to separate into parts, groups, sections, etc.

There is an infinite amount of divisions because that is the paradox.

What does 'infinite' mean here? And why do you think you can multiply D/(infinite) by it?
2. indefinitely or exceedingly great: infinite sums of money.

I can multiply D/(infinite) by infinite because its the same as adding D/(infinite) and infinite amount of times.

Do me the great honor of learning basic words such as sections and dividing.

If infinity/infinity does not equal one then I have found the other solution to the paradox. This would be that infinity is not well enough defined to solve a paradox involving infinite.

Perhaps this can help:

When dealing with points A and B of finite length, infinite segmentation is classically impossible.
To go even further, one can not segment infinity, as that would be inherently contradictory. What is one-half of infinity as opposed to infinity itself? (for example)

Anyway just some thoughts...

Another point I'd like to offer is that you can keep dividing mathematically forever, but there are practical limits in the real world... even the quantum level.

When one keeps mathematically dividing below certain levels it simply has no real meaning or effect on reality.

HallsofIvy
Homework Helper
If infinity/infinity does not equal one then I have found the other solution to the paradox. This would be that infinity is not well enough defined to solve a paradox involving infinite.
All you've really said here is that you do not understand the "paradox". I suspected that when I saw that this had been posted under "General Physics".

DaveC426913
Gold Member
Would you like me to define every word of my post to you?
No need to be snooty. And yes, you may need to define terms. That's what Zeno's Paradox is about - our nebulous grasp on meanings versus reality. We find that our words make a lot of assumptions, as you are finding you've done in your argument.

DaveC426913
Gold Member
If infinity/infinity does not equal one then I have found the other solution to the paradox. This would be that infinity is not well enough defined to solve a paradox involving infinite.
There is nothing in the paradox about infinity. Infinity only shows up in flawed attempts at a solution.

The solution to the paradox is to eliminate the infinite. The reason it took so long to be solved is because every other philosopher fell into the same infinity trap that you did.

rcgldr
Homework Helper
Isn't infinity a part of Calculus? There are methods used to calculate infinite sums. Integration to find area under a curve is the limit of the area as the number of segments approaches infinity.

In the case of two objects traveling at different but constant speeds, the ratio of the change in position versus the change in time is constant for each "interval", so the rate of closure and eventually passing by remains constant, even if the limit of the number of intervals approaches infinity while the size of the intervals approaches zero at the point where the faster object passes the slower one.

Another example, is a ball bouncing with a fixed percentage loss in energy on each bounce. The total time the ball bounces is fixed, but the number of times an abstract ball bounces is infinite. (In real life, eventually the bounces become smaller than the deformation of the ball, so it stops bouncing a bit sooner).

Hurkyl
Staff Emeritus
Gold Member
If infinity/infinity does not equal one then I have found the other solution to the paradox. This would be that infinity is not well enough defined to solve a paradox involving infinite.
I would agree that your concept of 'infinite' does not seem to be well-defined. However, mathematics gives it a more than adequate treatment (even to the satisfaction of philosophers!) -- so if you're serious about reflecting about 'infinite' things, you really need to study mathematics.

DaveC426913
Gold Member
Isn't infinity a part of Calculus? There are methods used to calculate infinite sums.
Absolutely. And therein will be an answer. But mere arithemetic - dividing and multiplying -i.e. mixing natural numbers and infinities - isn't it.
That's where the philosphers and the OP went awry.

rcgldr
Homework Helper
Absolutely. And therein will be an answer. But mere arithemetic - dividing and multiplying -i.e. mixing natural numbers and infinities - isn't it.
That's where the philosphers and the OP went awry.
True, but I'm not sure of the OP's math background. If you replace "infinite" with "n", and take the limit as n approaches infinite, then the OP had the right idea.

True, but I'm not sure of the OP's math background. If you replace "infinite" with "n", and take the limit as n approaches infinite, then the OP had the right idea.
The right idea to prove what? Was Zeno right or wrong in what he was trying to prove?

The OP seems to think Zeno was claiming any length is equal to an infinite length and any time duration is equal to an infinite duration of time. Or that that is true and thus negates what Zeno was trying to claim.
I cannot tell because the OP never says what Zeno was claiming to prove.

But neither does anyone else here which makes me think no one in this argument really knows what Zeno was trying to prove.

rcgldr
Homework Helper
But neither does anyone else here which makes me think no one in this argument really knows what Zeno was trying to prove.
This might help:

It's a paradox, which could mean that the author already knew the solution, but still considered it an interesting exercise for a potential reader.

Infinity/Infinity does not equal one in some cases... true. Such as 5x/x as x-> Infinity. However in this case, the n/n as n-> infinity does equal one.

Zeno claimed that movement wasn't possible because there was an infinite number of divisions to overcome. What I am saying is that because the number of devisions gets infinitely large, the time to cross those would be infinitely small. So these infinities would cancel.

If you can't figure out if I am proving or disproving the paradox perhaps it would be helpful to read the Paradox in full from the link Jeff provided. I was mistaken not have added that in my OP.

DaveC426913
Gold Member
Infinity/Infinity does not equal one in some cases... true. Such as 5x/x as x-> Infinity. However in this case, the n/n as n-> infinity does equal one..
Yes.

Zeno claimed that movement wasn't possible because there was an infinite number of divisions to overcome. What I am saying is that because the number of devisions gets infinitely large, the time to cross those would be infinitely small. So these infinities would cancel.
No.

Your assuming, and thus calculating, that infinity resides between two finite points.

This is not possible.

rcgldr
Homework Helper
Your assuming, and thus calculating, that infinity resides between two finite points.
Or that an infinite number of points exist between two finite points, similar to the fact that an infinite number of real numbers exist between any two real numbers.

Or that an infinite number of points exist between two finite points, similar to the fact that an infinite number of real numbers exist between any two real numbers.
Your comment seems out of the blue. How do you get "...an infinite number of points exist between two finite points," from $$\frac{\infty}{\infty}$$?

I think what you're trying to say is that as x approaches 0, t also approaches 0.

rcgldr
Homework Helper
Your assuming, and thus calculating, that infinity resides between two finite points.
Or that an infinite number of points exist between two finite points, similar to the fact that an infinite number of real numbers exist between any two real numbers.
Your comment seems out of the blue. How do you get "...an infinite number of points exist between two finite points," from $$\frac{\infty}{\infty}$$? I think what you're trying to say is that as x approaches 0, t also approaches 0.
No, just a response to the comment that infinity resides between two "finite" points. A better analogy to the OP's paradox is that an infinite number of events (the location where the faster object reaches a point where the slower object was at the the previous event), can exist within a finite distance and/or time.

Zenoman;
You and others are continuing to argue as if Zeno thought things like motion, finite points, distance, or things at distances were real things.
Did any of you look at the Wiki link Jeff gave? With other links there it does a reasonable job of explaining Zeno did not believe those were real.
And creating an infinity from something that does not exist proves nothing to Zeno.

The reduction to the absurd – was not the method used to falsify Zeno’s conclusions as some seem to indicate in this thread.
It was the method Zeno used to prove his point that the idea that any “thing” real moves at all was false.

Zeno’s argument was with the “pluralists” (pluralism at the time believed things really do move within a plenum similar to what Newton centuries later called absolute space and absolute time)

The point of Zeno’s arguments is he was used the logic of those that believe things do move “pluralists” to construct a paradox based on their own pluralist rules of movement that reduces to an absurd result; therefore refuting as absurd the idea that anything moves at all;
Thus supporting Zeno’s monism (what today we would call “Absolute Monism”) which believe that all is just one thing within which we and everything only exist as essentially an idea with nothing real actually moving.

As the Wiki info states in pure logic no one since then though the 20th Century has yet to falsify the Zeno Dichotomy Logic with out first assuming in principle that things move; And that in logic terms incorrectly “begs the question” posed in the first place.

Nothing in this thread certainly not the OP brings something new to resolve the issue as raised by Zeno.

Science seems to do just fine though by ignoring the Dichotomy and like paradoxes as meaningless sophist arguments. They quite literally were put forward by sophists, the problem is justifying with logic that sophist ideas are meaningless.

If you wish to resolve Zeno’s Dichotomy, it needs to be done in the context it was presented without “begging the question”.

Extra note:
Also since “Science”, (General Physics included) can be said to include as a principle foundation that Sophist agreements are not useful;
it is only logical that science cannot resolve a sophist problem.
That is this is more of a Logic and Philosophical issue; it might be better placed in the Philosophy Forum.

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DaveC426913
Gold Member
The point of Zeno’s arguments is he was used the logic of those that believe things do move “pluralists” to construct a paradox based on their own pluralist rules of movement that reduces to an absurd result; therefore refuting as absurd the idea that anything moves at all;
Thus supporting Zeno’s monism (what today we would call “Absolute Monism”) which believe that all is just one thing within which we and everything only exist as essentially an idea with nothing real actually moving.
While I don't doubt that you're making a valid point, I cannot understand what you are saying, especially in the above. It is grammatically so poorly-formed as to be unintelligible.